Op-Amps

For an analog electronics exam, you'll need formulas covering key topics like circuit analysis, semiconductor devices, amplifiers, filters, and feedback systems. Below is a categorized list of essential formulas, their applications, and derivations where useful.

1. Basic Circuit Analysis

Ohm’s Law:

$$ V=IR $$

Application: Used for solving simple resistor networks.

Kirchhoff’s Laws:

KVL (Kirchhoff’s Voltage Law): The sum of voltages in a closed loop is zero. $$ ∑V=0 $$
KCL (Kirchhoff’s Current Law): The sum of currents entering a node equals the sum of currents leaving.

∑I=0 $$

Application: Used for analyzing circuits with multiple loops and nodes.

Thevenin’s & Norton’s Theorems:

Thevenin Equivalent Resistance: $$ Rth=VocIscR_{\text{th}} = \frac{V_{\text{oc}}}{I_{\text{sc}}}Rth=IscVoc $$
Norton Equivalent Current:

IN=VocRthI_{\text{N}} = \frac{V_{\text{oc}}}{R_{\text{th}}}IN=RthVoc

Application: Simplifies complex circuits into a single voltage or current source with equivalent resistance.

2. Diodes and Rectifiers

Diode Current Equation (Shockley Equation):

$$ ID=IS(eVDnVT−1)I_D = I_S \left( e^{\frac{V_D}{nV_T}} -1 \right)ID=IS(enVTVD−1) $$ Where:

ISI_SIS = Reverse saturation current $$V_T = Thermal voltage (≈25mV at 300K)$$
nnn = Ideality factor (1 for Si, 2 for Ge)
Application: Determines the current flowing through a diode for a given voltage.

Rectifier Circuits:

Half-wave rectifier output voltage (ideal): $$ Vdc=VmπV_{\text{dc}} = \frac{V_m}{\pi}Vdc=πVm $$
Full-wave rectifier output voltage (ideal): $$ Vdc=2VmπV_{\text{dc}} = \frac{2V_m}{\pi}Vdc=π2Vm $$
Ripple voltage for capacitor filter: $$ Vripple=IfCV_{\text{ripple}} = \frac{I}{fC}Vripple=fCI $$
Application: Used for AC-to-DC conversion.

3. Bipolar Junction Transistor (BJT)

DC Analysis of BJT:

Base current: $$ IB=VB−VBERBI_B = \frac{V_B - V_{BE}}{R_B}IB=RBVB−VBE $$
Collector current (active region): $$ IC=βIBI_C = \beta I_BIC=βIB $$
Emitter current: $$ IE=IC+IB=ICαI_E = I_C + I_B = \frac{I_C}{\alpha}IE=IC+IB=αIC $$ Where: $$ α=ββ+1\alpha = \frac{\beta}{\beta +1}α=β+1β $$
Application: Determines operating point in amplifier design.

AC Analysis (Small-Signal Model):

Transconductance: $$ gm=ICVTg_m = \frac{I_C}{V_T}gm=VTIC $$
Input resistance (common emitter amplifier): $$ rπ=βgmr_\pi = \frac{\beta}{g_m}rπ=gmβ $$
Voltage gain (CE amplifier): $$ Av=−βRCrπA_v = -\beta \frac{R_C}{r_\pi}Av=−βrπRC $$
Application: Used for designing BJT amplifiers.

4. Field Effect Transistor (FET & MOSFET)

DC Analysis of MOSFET (Saturation Region):

$$ ID=12μnCoxWL(VGS−Vth)2I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2ID=21μnCoxLW(VGS−Vth)2 $$

Application: Determines MOSFET operating point.

Small-Signal Model of MOSFET:

Transconductance:

gm=2IDVGS−Vthg_m = \frac{2I_D}{V_{GS} - V_{th}}gm=VGS−Vth2ID

Voltage gain:

Av=−gmRDA_v = -g_m R_DAv=−gmRD

Application: Used in MOSFET amplifier design.

5. Amplifiers

Common-Emitter Voltage Gain:

Av=−βRCrπA_v = -\beta \frac{R_C}{r_\pi}Av=−βrπRC

Derivation: Using small-signal model, Vout=−ICRCV_{\text{out}} = -I_C R_CVout=−ICRC, and IC=βIBI_C = \beta I_BIC=βIB, substituting IB=Vin/rπI_B = V_{\text{in}}/r_\piIB=Vin/rπ gives:

Av=−βRCrπA_v = -\beta \frac{R_C}{r_\pi}Av=−βrπRC

Gain-Bandwidth Product (for Op-Amps):

Gain×Bandwidth=constant\text{Gain} \times \text{Bandwidth} = \text{constant}Gain×Bandwidth=constant

Application: Used to determine op-amp stability and bandwidth.

6. Feedback and Stability

Closed-Loop Gain (Negative Feedback):

Af=A1+AβA_f = \frac{A}{1 + A \beta}Af=1+AβA

Where:

AAA = Open-loop gain
β\betaβ = Feedback factor
Application: Controls amplifier gain and bandwidth.

Bode Stability Criterion:

Phase margin:

PM=180∘+∠Loop Gain\text{PM} = 180^\circ + \angle \text{Loop Gain}PM=180∘+∠Loop Gain

Application: Ensures amplifier stability.

7. Filters and Frequency Response

First-Order Low-Pass Filter:

H(jω)=11+jωωcH(j\omega) = \frac{1}{1 + j\frac{\omega}{\omega_c}}H(jω)=1+jωcω1

Cutoff frequency:

ωc=1RC\omega_c = \frac{1}{RC}ωc=RC1

First-Order High-Pass Filter:

H(jω)=jωωc1+jωωcH(j\omega) = \frac{j\frac{\omega}{\omega_c}}{1 + j\frac{\omega}{\omega_c}}H(jω)=1+jωcωjωcω

Application: Designs active/passive filters.

8. Oscillators

Barkhausen Criterion:

Aβ=1,∠Aβ=0∘ or 360∘A \beta = 1, \quad \angle A\beta = 0^\circ \text{ or } 360^\circAβ=1,∠Aβ=0∘ or 360∘

Application: Ensures sustained oscillations in circuits.

Wien Bridge Oscillator Frequency:

f=12πRCf = \frac{1}{2\pi RC}f=2πRC1

Application: Used for sine wave generation.

9. Power Electronics

DC-DC Converter Efficiency:

η=PoutPin×100%\eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100%η=PinPout×100%

Application: Analyzes power conversion efficiency.

Buck Converter Output Voltage:

Vout=DVinV_{\text{out}} = D V_{\text{in}}Vout=DVin

Application: Designs step-down regulators.

Final Tips for Application:

Understand Derivations – Many formulas come from circuit laws and small-signal models.
Use SPICE/LTspice Simulations – Verify results for circuit performance.
Memorize Key Expressions – Focus on amplifier gains, filter cutoffs, and transistor models.
Practice Complex Problems – Solve past exam questions to solidify concepts.

Would you like step-by-step derivations for any specific formula?

You said:

Are all these the formules i need??if there are more please provide them

3/3

ChatGPT said:

The formulas I provided cover the core topics in analog electronics, but depending on the depth of your exam, you may need more advanced or specific formulas. Below, I expand on additional topics and provide more formulas that could be useful.

10. Resonance and Frequency Response

Series RLC Circuit:

Resonant Frequency:

fr=12πLCf_r = \frac{1}{2\pi \sqrt{LC}}fr=2πLC1

Quality Factor:

Q=1RLCQ = \frac{1}{R} \sqrt{\frac{L}{C}}Q=R1CL

Bandwidth:

BW=frQBW = \frac{f_r}{Q}BW=Qfr

Application: Used in tuned amplifiers and oscillators.

11. Operational Amplifiers (Op-Amps)

Ideal Op-Amp Characteristics:

Infinite open-loop gain (A→∞A \to \inftyA→∞)
Infinite input impedance (Rin→∞R_{\text{in}} \to \inftyRin→∞)
Zero output impedance (Rout=0R_{\text{out}} = 0Rout=0)

Common Op-Amp Configurations:

Inverting Amplifier:

Av=−RfRinA_v = -\frac{R_f}{R_{in}}Av=−RinRf

Non-Inverting Amplifier:

Av=1+RfRinA_v = 1 + \frac{R_f}{R_{in}}Av=1+RinRf

Voltage Follower (Buffer):

Av=1A_v = 1Av=1

Differentiator:

Vout=−RCdVindtV_{\text{out}} = -RC \frac{dV_{\text{in}}}{dt}Vout=−RCdtdVin

Integrator:

Vout=−1RC∫VindtV_{\text{out}} = -\frac{1}{RC} \int V_{\text{in}} dtVout=−RC1∫Vindt

Application: Used in signal processing, active filters, and control systems.

12. Feedback in Amplifiers

General Negative Feedback Gain Equation:

Af=A1+AβA_f = \frac{A}{1 + A\beta}Af=1+AβA

Effect of Negative Feedback:
- Reduces gain variation
- Increases bandwidth
- Improves linearity
- Reduces distortion
Application: Essential in designing stable amplifiers.

13. Power Amplifiers

Efficiency of Power Amplifiers:

Class A:

ηmax=25%\eta_{\text{max}} = 25%ηmax=25%

Class B:

ηmax=78.5%\eta_{\text{max}} = 78.5%ηmax=78.5%

Class AB: η\etaη is between Class A and B
Class C:

ηmax>80%\eta_{\text{max}} > 80%ηmax>80%

Application: Determines power efficiency in audio and RF applications.

14. Noise in Electronic Circuits

Thermal Noise (Johnson-Nyquist Noise):

Vn=4kBTRBV_n = \sqrt{4 k_B T R B}Vn=4kBTRB

Where:

kBk_BkB = Boltzmann’s constant (1.38×10−23J/K1.38 \times 10^{-23} J/K1.38×10−23J/K)
TTT = Temperature in Kelvin
RRR = Resistance (Ω\OmegaΩ)
BBB = Bandwidth (Hz)
Application: Important in low-noise amplifier design.

15. Voltage Regulation

Zener Diode Regulation:

$$ Vout=VZV_{\text{out}} = V_ZVout=VZ $$

Linear Regulator (Basic Series Regulator):

$$ Vout=Vin−VdropV_{\text{out}} = V_{\text{in}} - V_{\text{drop}}Vout=Vin−Vdrop $$

Application: Provides stable DC voltage for circuits.

16. Schmitt Triggers and Comparators

Threshold Voltages for a Schmitt Trigger (Hysteresis Effect):

Upper threshold: $$ VTH+=R2R1+R2VccV_{TH+} = \frac{R_2}{R_1 + R_2} V_{\text{cc}}VTH+=R1+R2R2Vcc $$
Lower threshold: $$ VTH−=R2R1+R2VeeV_{TH-} = \frac{R_2}{R_1 + R_2} V_{\text{ee}}VTH−=R1+R2R2Vee $$
Application: Used to eliminate noise in digital circuits.

17. Analog Filters (Higher Order)

Second-Order Low-Pass Filter (Sallen-Key):

$$ H(s)=1s2+sQ+1H(s) = \frac{1}{s^2 + \frac{s}{Q} + 1}H(s)=s2+Qs+11 $$ Where: $$ Q=12Q = \frac{1}{\sqrt{2}}Q=21 $$ For Butterworth filter (maximally flat response).

Application: Used in audio processing and signal conditioning.

18. Phase-Locked Loops (PLL)

Basic PLL Equation:

$$ fout=Nfreff_{\text{out}} = N f_{\text{ref}}fout=Nfref $$

Application: Used in frequency synthesis and clock recovery.

Application: Used in RF and microwave engineering.

20. Miscellaneous Formulas

Decibel (dB) Calculations:

Voltage gain in dB: $$ Av(dB)=20log⁡(VoutVin)A_v (dB) = 20 \log \left( \frac{V_{\text{out}}}{V_{\text{in}}} \right)Av(dB)=20log(VinVout) $$
Power gain in dB: $$ Ap(dB)=10log⁡(PoutPin)A_p (dB) = 10 \log \left( \frac{P_{\text{out}}}{P_{\text{in}}} \right)Ap(dB)=10log(PinPout) $$
Noise Figure: $$ NF(dB)=10log⁡(SNRinSNRout)NF (dB) = 10 \log \left( \frac{\text{SNR}{\text{in}}}{\text{SNR}{\text{out}}} \right)NF(dB)=10log(SNRoutSNRin) $$
Application: Used in communication and audio systems.

Final Notes

These formulas cover most topics you are likely to encounter in an analog electronics exam. However, some exams may focus on particular areas like:

RF design (Smith charts, impedance matching)
Control systems (Bode plots, Nyquist stability)
Power electronics (SMPS, inverter design)